Penny S. Reynolds · 2023
Ordinal score data are commonly used in animal-based studies, surveys and questionnaires, and observer agreem ent studies. Ordinal data consist of counts obtained for two or more categories with a meaningful order or ranking (for example, ‘very poor’ to ‘very good’). Unlike continuous data, basic arithmetic operations cannot be performed on ordinal data, and descriptive statistics such as means do not have a sensible interpretation. Sample size approximation methods are based on the proportion of events expected for each category and the anticipated effect size, expressed as an odds ratio. Sample size increases substantially when proportions differ across categories, especially when one category predominates over the others. Rater agreement studies are a special case. Here two or more observers assess the same specimens or items with the goal of assessing measurement reproducibility and repeatability. Sample size is based on Cohen's or Fleiss’ kappa coefficients and the expected number of discordant ratings between observers.
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Alan Agresti · Wiley StatsRef: Statistics Reference Online · 2015
Abstract An ordinal variable is one that has a natural ordering of its possible values, but for which the distances between the values are undefined. For simple bivariate analyses, measures of association such as generalizations of Kendall's tau summarize the strength of positive or negative trend in the data by evaluating relative frequencies of concordant and discordant pairs of observations. Log‐linear models with scores for the categories can describe the structure of the joint distribution. Generalizations of the binary logistic regression model describe the dependence of an ordinal response variable on a set of qualitative and/or quantitative explanatory variables. Most commonly used is the cumulative logit model , which applies the logit link function to the set of cumulative probabilities. The proportional odds version of the model assumes the same effect on the logit of each cumulative probability. Extensions of the model apply to correlated multivariate ordinal responses; such extensions often occur in longitudinal studies. These extensions include marginal models , which simultaneously describe effects of explanatory variables on each marginal response, and generalized linear mixed models with random effects .
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Nairanjana Dasgupta, Jillian Morrison · 2024
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Rune Haubo Bojesen Christensen · 2010
Implementation of cumulative link (mixed) models also known as ordered regression models, proportional odds models, proportional hazards models for grouped survival times and ordered logit/probit/... models. Estimation is via maximum likelihood and mixed models are fitted with the Laplace approximation and adaptive Gauss-Hermite quadrature. Multiple random effect terms are allowed and they may be nested, crossed or partially nested/crossed. Restrictions of symmetry and equidistance can be imposed on the thresholds (cut-points/intercepts). Standard model methods are available (summary, anova, drop-methods, step, confint, predict etc.) in addition to profile methods and slice methods for visualizing the likelihood function and checking convergence.
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Norman Cliff · PsycEXTRA Dataset · 1994
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Nairanjana Dasgupta, Jillian Morrison · 2024
This book is a step-by-step data story for analyzing ordinal data from start to finish. The book is for researchers, statisticians and scientists who are working with datasets where the response is ordinal. This type of data is common in many disciplines, not just in surveys (as is often thought). For example, in the biological sciences, there is an interest in understanding and predicting the (growth) stage (of a plant or animal) based on a multitude of factors. Likewise, ordinal data is common in environmental sciences (for example, stage of a storm), chemical sciences (for example, type of reaction), physical sciences (for example, stage of damage when force is applied), medical sciences (for example, degree of pain) and social sciences (for example, demographic factors like social status categorized in brackets). There has been no complete text about how to model an ordinal response as a function of multiple numerical and categorical predictors. There has always been a reluctance and reticence towards ordinal data as it lies in a no-man’s land between numerical and categorical data. Examples from health sciences are used to illustrate in detail the process of how to analyze ordinal data, from exploratory analysis to modeling, to inference and diagnostics. This book also shows how Likert-type analysis is often used incorrectly and discusses the reason behind it. Similarly, it discusses the methods related to Structural Equations and talks about appropriate uses of this class of methods. The text is meant to serve as a reference book and to be a “how-to” resource along with the “why” and “when” for modeling ordinal data. Key Features: Includes applications of the statistical theory Includes illustrated examples with the associated R and SAS code Discusses the key differences between the different methods that are used for ordinal data analysis Bridges the gap between methods for ordinal data analysis used in different disciplines
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Valen E. Johnson, James H. Albert · Statistics for social and behavioral sciences · 1999
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Mary Kathryn Cowles, Valen E. Johnson, James H. Albert · Journal of the American Statistical Association · 2000
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Xingye Qiao · Wiley Interdisciplinary Reviews Computational Statistics · 2015
Classification is an important topic in statistical learning. The goal of classification is to build a predictive model from the training dataset for the class label of an observation. It is commonly assumed that the class labels are unordered. However, in many real applications, there exists an intrinsic ordinal relation between the class labels. Examples of these include cancer patients grouped in early, mediocre, and terminal stages, customers grouped into low, middle, and high credit levels, and experimental subjects enriched with different amounts of bacterial. In this article, we focus on the classification problem for ordinal data and introduce the theoretical setup of the problem. We review both traditional and modern methods in learning ordinal data. In particular, we emphasize the trade‐off between model flexibility and interpretability. Lastly, we discuss some issues regarding ordinal data learning, including an appropriate loss function for this problem. WIREs Comput Stat 2015, 7:341–346. doi: 10.1002/wics.1357 This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
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Gerd Stumme · 2019
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Pier Alda Ferrari, Alessandro Barbiero · Multivariate Behavioral Research · 2012
The increasing use of ordinal variables in different fields has led to the introduction of new statistical methods for their analysis. The performance of these methods needs to be investigated under a number of experimental conditions. Procedures to simulate from ordinal variables are then required. In this article, we deal with simulation from multivariate ordinal random variables. We propose a new procedure for generating samples from ordinal random variables with a prespecified correlation matrix and marginal distributions. Its features are examined and compared with those of its main competitors. A software implementation in R is also provided along with examples of its application.
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Muhammad Ridho Febrian Saputra · 2019
Bagian lain dari data kontinum adalah data ordinal. Data ini, selain memiliki nama (atribut), juga memiliki peringkat atau urutan. Angka yang diberikan mengandung tingkatan. Ia digunakan untuk mengurutkan objek dari yang paling rendah sampai yang paling tinggi, atau sebaliknya. Ukuran ini tidak memberikan nilai absolut terhadap objek, tetapi hanya memberikan peringkat saja. Jika kita memiliki sebuah set objek yang dinomori, dari 1 sampai n, misalnya peringkat 1, 2, 3, 4, 5 dan seterusnya, bila dinyatakan dalam skala, maka jarak antara data yang satu dengan lainnya tidak sama. Ia akan memiliki urutan mulai dari yang paling tinggi sampai paling rendah. Atau paling baik sampai ke yang paling buruk.
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Xiaoting Sun · ResearchSpace (University of Auckland) · 2014
Full text is available to authenticated members of The University of Auckland only.
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National Cancer Institute · Definitions · 2020
Ope n Pe
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Frank Cowell, Emmanuel Flachaire · Economica · 2017
The standard theory of inequality measurement assumes that the equalisand is a cardinal quantity, with known cardinalization. However, one often needs to make inequality comparisons where either the cardinalization is unknown or the underlying data are categorical. We propose an alternative approach to inequality analysis that is rigorous, has a natural interpretation, and embeds both the ordinal data problem and the well‐known cardinal data problem. We show how the approach can be applied to the inequality of happiness and of health status.
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Ikechukwu, Blessing Ukamaka · Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT) · 2024
Several clustering methods have been developed for continuous multivariate data. However, these methods are often applied to analyse and cluster ordinal data as if they possess metric properties, thereby overlooking their ordinal nature. This study focuses on model-based clustering methods based on parameterized finite Gaussian mixture modeling, which utilize Expectation-Maximization (EM) algorithms to identify clusters. Using several selected datasets, the recovery ability of these types of clustering methods is investigated when the ratio scale of the data, if not originally ordinal, is changed to an ordinal scale. The EM algorithm clustering method for ordinal data considered was the pairwise likelihood approach developed by Ranalli and Rocci (2016). This EM algorithm approach was used to estimate the parameters of the mixture model. For a standard URV (Underlying Response Variable) approach, it is assumed that the ordinal variables are generated by a discretization of underlying multivariate normal variables. In this work, an extension of the URV approach was applied by taking a mixture of multivariate normal distributions. The pairwise EM algorithm was compared with a model-based clustering (EM algorithm) method, the Mclust function, developed in R. When the data is originally ordinal, the probit ordinal model was used in generating the underlying continuous data, which is assumed to be normally distributed, before applying the URV method. Three datasets were considered: The first dataset is the Iris dataset with 150 observations and 4 numerical variables. The second dataset is a public dataset on maternal health with 1014 observations which includes 7 measurements, 6 continuous and 1 ordinal. The third dataset is on the risk of dementia among patients with HIV/AIDS. Consists of 255 observations with 4 ordinal variables. The continuous variables in the Iris and the maternal health risk data were discretized to analyse them as ordinal data. Since the underlying latent mixture is known, the thresholds used for discretization of the latent variables, in the Pairwise EM algorithm, were those that maximized the adjusted rand index (ARI). The URV extension method recovered the cluster structure of the Iris data with an ARI of 0.922 and that of the maternal data with an ARI of 0.3. Applying Mclust on the datasets ARI of 0.90 and 0.15 were obtained for the Iris and maternal health datasets respectively. For the dementia data, a 4-component model was selected as the best model. The URV extension method can recover the cluster structure of the data even when it is applied to incomplete data and also to data that do not completely represent the original data (due to discretization). The three applications show that changing measurement scales from continuous (original data) to ordinal (via discretization) using the URV method, can enhance the recovery ability of model-based clustering (EM algorithm) methods.
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Nancy Organ · 2024
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Justice Nii-Ayitey · Washington State University · 2025
The problem we aim to address in this dissertation is to develop a more straightforward method for examining correlation-related diagnostics in ordinal data. Analyzing data that falls between numerical and categorical types can be challenging, particularly when it involves ordinal variables. One major challenge is the limited availability of diagnostics for evaluating the performance of ordinal regression models. Most often, the diagnostics performed for ordinal regression are the proportional odds assumption and the goodness-of-fit test. The proportional odds assumption evaluates whether the relationship between the predictors and the odds of being in a higher versus a lower category is consistent across thresholds, while the goodness-of-fit test assesses whether the model fits the data better than a null model. While these are somewhat easily available in software, the problem of dependence is not. The two main issues related to correlation in ordinal data are: multicollinearity, when the predictors are highly correlated, and autocorrelation or familial dependence, when the responses are correlated with each other. While diagnostics for these are easily available in the General Linear Model (GLM), this is not the case for ordinal data. The approaches that exist for ordinal data are theoretically and computationally cumbersome and time-consuming. For multicollinearity, we can directly examine the structure of the design matrix and predictors, and consider eigenvalues and condition numbers, as in GLM. However, for ordinal data, our estimates are based on maximum likelihood estimates, and thus a direct examination of the design matrix is not necessarily the best approach. Similarly, we could consider generalized estimating equations or Binary Dynamic Logit for Correlated Ordinal (BDLCO) method for modeling the dependence structure, which are computationally intensive. We aim to determine when BDLCO is needed versus using traditional methods for ordinal data, assuming proportional odds. In this dissertation, our primary objective is to develop a “quick and dirty” method to determine whether multicollinearity is indeed an issue and whether methods like ridge or lasso are warranted. In the same way, we try to quickly diagnose "dependence issues". If dependence is detected, we would suggest using a method like BDLCO. Our approach is to utilize our ordinal response and its underlying relationship with predictors through the slope estimates to estimate the underlying latent response as a function of the predictors, and then use readily available methods to perform quick checks for multicollinearity or autocorrelation. Essentially, we show that the slope estimates from the ordinal response can aid in estimating the slope estimates for the underlying latent variable in an Approximate Latent Linear Model (ALLM). The idea is that once we estimate such a continuous response, then we can use the multivariate normal distribution of the approximate latent linear response to understand autocorrelation and multicollinearity. Thus, we can utilize standard diagnostic techniques developed within the GLM framework, such as the Durbin-Watson test and variance inflation factors, to address issues of autocorrelation and multicollinearity. Our simulation results show that this procedure gives us a quick way to understand and diagnose problems that are theoretically and computationally difficult in a much faster way. We use our method on a data example, which helps to make decisions about ordinal data in the presence of dependence.
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Euis Sartika · Sigma-Mu · 1970
Dalam analisis Multivariat, pengolahan data terkadang mengharuskan data berskala interval / metrik. Apabila data yang dihadapi berskala ordinal, sebaiknya digunakan analisis multivariat nonparametrik. Namun, analisis ini mempunyai banyak kesulitan. Karena software yang mampu mengakomodasi teknik-teknik seperti ini masih kurang. Dalam Psikometrika, analisis yang digunakan untuk mengatasi masalah ini dikenal dengan metode interval berurutan (succesive interval). Akan tetapi, teknik ini mengasumsikan data populasi berdistribusi normal. Data ordinal tidak memiliki makna jarak antarkategori. Hal tersebut dapat diubah dengan mentranformasi data ordinal menjadi interval. Transformasi data ordinal menjadi interval pada dasarnya dilakukan dengan mengubah proporsi kumulatif tiap peubah pada kategori menjadi nilai kurva normal bakunya
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Francesco Amato, Julien Jacques · HAL (Le Centre pour la Communication Scientifique Directe) · 2022
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